Certification of non-classicality in all links of a photonic star network without assuming quantum mechanics

Networks composed of independent sources of entangled particles that connect distant users are a rapidly developing quantum technology and an increasingly promising test-bed for fundamental physics. Here we address the certification of their post-classical properties through demonstrations of full network nonlocality. Full network nonlocality goes beyond standard nonlocality in networks by falsifying any model in which at least one source is classical, even if all the other sources are limited only by the no-signaling principle. We report on the observation of full network nonlocality in a star-shaped network featuring three independent sources of photonic qubits and joint three-qubit entanglement-swapping measurements. Our results demonstrate that experimental observation of full network nonlocality beyond the bilocal scenario is possible with current technology.


Supplementary Note 1 -Experimental data
Below we collect the experimental counts of coincidence events in the experiment, upon successful measurement of b = 0 in the central node.

Supplementary Note 2 -Experimental source independence
In order to estimate the degree of independence between the sources in the experiment, we calculate the mutual information (MI) between each pair of wing parties with different in- is the Shannon entropy. For completely statistically independent parties, the mutual information should be zero. In the experiment, we set the three HWPs in the central node to 0 • in order to project the three photons into the computational basis, and the three wing parties are switched between the A 0 and A 1 measurements randomly. By summing over all the outcomes of the central node, we obtain the joint and individual outcome probabilities of the wing parties. The calculated mutual information for different inputs is shown in Supplementary Table 1.

Supplementary Note 3 -FNN in the bilocal scenario
By removing one of the sources and one of the PBSs in the central party, we also test the full network nonlocality witnesses for the bilocal scenario. The experimental setup is shown in Supplementary Figure 1, where two sources are used to link the three nodes.
As in the three-star case, Alice and Charlie both can perform two binary-outcome measurements, i.e., x, z, a, c ∈ {0, 1}. However, now the central party Bob performs a fixed measurement with three possible outcomes, so b ∈ {0, 1, 2}. In this scenario, all non-FNN Supplementary Values close to zero verify the independence of the respective sources. Each row corresponds to a fixed choice of measurements for the pair of parties.
In the experiment, both SPDC sources produce entangled photon pairs in the singlet state 1 √ 2 (|HV ⟩ − |V H⟩). The measurement observables for Alice and Charlie are A 0 = σ x , A 1 = σ z , C 0 = σz+σx √ 2 and C 1 = σz−σx √ 2 . At the central node, Bob performs a partial Bell state measurement by overlapping the two input photons on a PBS and detected in the |±⟩ basis. In order to detect three outcomes, Bob implements pseudo-numberresolving detectors which consist of a 22.5 • HWP, a PBS and two fiber-coupled avalanche photodiodes (APDs) for each of his four output ports. Thus the partial BSM can resolve the two states |ϕ ± ⟩ = (|HH⟩ + |V V ⟩)/ √ 2, and the unresolved events are associated to the outcome b = 2. The resulting theoretical distribution leads to the quantum violations Supplementary Figure 1: Experimental setup for the FNN experiment in the bilocal scenario. The ultraviolet pulse is divided into two parallel beams and the relative phases between them are erased by two randomly rotated optical windows. Ordinary photons from two EPR sources are sent to the two end nodes respectively. Extraordinary photons are sent to the central node and measured by a partial Bell state measurement device. PBS, polarisation beam splitter; HWP, half-wave plate; IF, interference filter; QRNG, quantum random number generator; BBO, beta barium borate.
We use 1-and 3-nm bandwidth filters for, respectively, the e-and o-polarised photons, and use a pump power of 30 mW for each source. The two-photon counting rate is about 9000 Hz and the four-photon coincidence rate is about 0.8 Hz. As in the experiment described in the main text, in order to improve the independence of the sources we insert a randomly rotated glass slice before each source. After collecting data for 10 000 seconds for each setting, the experimentally measured results are R C-NS = 3.4966 ± 0.0238 and R NS-C = 3.4166 ± 0.0237, violating the non-FNN bounds by more than 17 standard deviations. The raw data is shown below. We note that the raw data must be postprocessed for the events with b = 2 due to the limited photon-resolving capabilities of the setup. Namely, when the input to Bob's measurement station is |ψ ± ⟩ = (|01⟩ ± |10⟩)/ √ 2, the output from the top PBS is two indistinguishable photons either in the left part or in the right part of the station. These two photons have a 50% chance of arriving to